Engineering
Palmgren-Miner damage index and a Basquin S-N life example
Sum cycle ratios in a linear fatigue model and explore a separate Basquin cycle-life calculation with an explicit reversals convention.
A linear damage index, not a safety verdict
Miner’s index sums applied cycles divided by the corresponding fitted constant-amplitude lives. The life data must match stress amplitude, mean stress, material, environment and the endpoint used to define fatigue life. Fractional cycle counts from cycle counting are allowed.
D=1 is the nominal endpoint of this linear rule, not proof that a crack appears at exactly that value. Load sequence effects, interactions and scatter are omitted. There is no universal aerospace threshold of 0.70.
The complement MAX(0,1−D) is unused model damage allowance; it is not a measured fraction of remaining calendar or service life. Converting it to future cycles requires an assumed future load spectrum. A below-threshold result does not establish component safety.
| Spectrum Stress Block | Applied Cycles n_i | S-N Life N_i | Damage Fraction n_i / N_i |
|---|---|---|---|
| Block 1 (Severe Gusts) | 40,000 | 100,000 | 0.4000 damage contribution |
| Block 2 (Maneuver Loads) | 150,000 | 500,000 | 0.3000 damage contribution |
| Block 3 (Cruise Ripple) | 200,000 | 1,000,000 | 0.2000 damage contribution |
| Total Miner Damage D | SUM(n_i / N_i) | — | 0.9000 (0.1000 model allowance) |
Palmgren-Miner Cumulative Fatigue Damage Index D = SUM(n_i / N_i)
Sums cycle-ratio fractions across all stress-amplitude bins.
=IF(COUNT(B2:B7)<>6,"Invalid inputs",IF(OR(B2<0,B4<0,B6<0,B3<=0,B5<=0,B7<=0),"Invalid inputs",ROUND(B2/B3+B4/B5+B6/B7,4)))Compare with the nominal model threshold
Compare the unrounded damage sum with one. This message is not a safe/fail engineering assessment.
=IF(COUNT(B2:B7)<>6,"Invalid inputs",IF(OR(B2<0,B4<0,B6<0,B3<=0,B5<=0,B7<=0),"Invalid inputs",IF((B2/B3+B4/B5+B6/B7)>=1,"Nominal model threshold reached","Below nominal model threshold")))Unused model damage allowance
The nonnegative complement is a model quantity, not an observed remaining service lifetime.
=IF(COUNT(B2:B7)<>6,"Invalid inputs",IF(OR(B2<0,B4<0,B6<0,B3<=0,B5<=0,B7<=0),"Invalid inputs",MAX(0,ROUND(1-(B2/B3+B4/B5+B6/B7),4))))Basquin life: cycles versus reversals
For the convention amplitude = strength coefficient × (2N)^b, invert to N=0.5×(amplitude/coefficient)^(1/b). Both stresses use MPa; b is negative. These invented coefficients demonstrate arithmetic and are not material design data.
The example is separate from the supplied Miner block lives. Do not extrapolate a fitted S-N curve outside its validated range or assume it captures an endurance limit.
=IF(COUNT(E2:E4)<>3,"Invalid inputs",IF(OR(E2<=0,E3<=0,E4>=0),"Invalid inputs",0.5*(E2/E3)^(1/E4)))Try the example
Use the original values or change them to test your understanding.